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A real-valued function  $  f $
defined on a certain set  $  E $
of real numbers such that the condition

$$ 
x  ^  \prime   <  x  ^ {\prime\prime} ,\ \ 
x  ^  \prime  , x  ^ {\prime\prime}  \in  E
$$

implies

$$ 
f ( x  ^  \prime  )  <  f ( x  ^ {\prime\prime} ).
$$

Such functions are sometimes called strictly increasing functions, the term  "increasing functions"  being reserved for functions which, for such given  $  x  ^  \prime  $
and  $  x  ^ {\prime\prime} $, 
merely satisfy the condition

$$ 
f ( x  ^  \prime  )  \leq   f ( x  ^ {\prime\prime} )
$$

(non-decreasing functions). The inverse function of any strictly increasing function is single-valued and is also strictly increasing. If  $  x _ {0} $
is a right-sided (or left-sided) limit point of the set  $  E $(
cf. [[Limit point of a set|Limit point of a set]]), if  $  f $
is a non-decreasing function and if the set  $  A = \{ {y } : {y = f( x),  x > x _ {0} ,  x \in E } \} $
is bounded from below — or if  $  \{ {y } : {y = f( x),  x < x _ {0} ,  x \in E } \} $
is bounded from above — then, as  $  x \rightarrow x _ {0} + $(
or, correspondingly, as  $  x \rightarrow x _ {0} - $), 
$  x \in E $, 
the values  $  f( x) $
will have a finite limit; if the set is not bounded from below (or, correspondingly, from above), the values  $  f ( x) $
have an infinite limit equal to  $  - \infty $(
or, correspondingly, to  $  + \infty $).

====Comments====
If  $  f $
is non-decreasing on  $  E $
and  $  x _ {0} \in E $, 
then the set  $  A $
referred to above is automatically bounded from below by  $  f ( x _ {0} ) $, 
unless it is empty. If, in addition,  $  x _ {0} $
is a limit point of  $  \{ {x \in E } : {x > x _ {0} } \} $, 
then the right-hand limit of  $  f $
at  $  x _ {0} $
is simply the infimum of  $  A $:

$$ 
\lim\limits _ {x \downarrow x _ {0} }  f ( x)  =  \inf  A .
$$

Similar remarks hold for left-hand limits.